I think I need pix to fully understand what it is that you've messed up. At this point it sounds like it's foobarred and you need to replace the HD or Mobo (or use a different IDE connector if u have one), whichever end we're talking about.

It is possible to wick up the solder with some solder wick however, might be able to save you from a short, but not positive depending on the severity of how much u coated the other pins.

If one of the pins came off i think your buggered. Solder will have made it worse, but u can remove it with a type of braid. Only problem is its dam hard to do, and i have no idea where to get the braid.

Some slim solder wick might do the job, but it's all going to depend on how much extra solder you got in there and if enough can be picked up so that no traces are connecting the pins when u get done. It's going to be a long and frustrating process if you give this method a try. You also risk bending/breaking more pins whilst trying to get the wick down through all of the pins. SATA FTW!